An Introduction to Representation Theory of Finite Groups Pooja Singla Ben-Gurion University of the Negev Be’er Sheva Israel February 28, 2011 Pooja Singla (BGU) Representation Theory February 28, 2011 1 / 37
An Introduction to Representation Theory of FiniteGroups
Pooja Singla
Ben-Gurion University of the NegevBe’er Sheva
Israel
February 28, 2011
Pooja Singla (BGU) Representation Theory February 28, 2011 1 / 37
History
The groups discussed were mainly
Symmteric groups
Z/nZ and (Z/nZ)?
GLn(C)
Cayley (1894) gave definition of abstract group
(Group) A set G is called group if there exists an operation? : G × G → G such that(1) ? is associative.(2)There exists an element 1 ∈ G such that 1a = a1 = a for all a ∈ G .(3) For every a ∈ G there exists a unique b ∈ G such thatab = ba = 1.
Pooja Singla (BGU) Representation Theory February 28, 2011 2 / 37
Simple Groups - Group having no proper normal subgroups.
Question
What are all the finite simple groups?
To answer this question various tools were discussed andRepresentation theory of finite groups is one of these.
Other motivation of representation theory comes from the study of groupactions.
Pooja Singla (BGU) Representation Theory February 28, 2011 3 / 37
Basic Definitions
G - Always finite group.
Definition
A representation of G is a homomorphism from G to the set ofautomorphisms of a finite dimensional complex vector space V , i.e.
φ : G 7→ GL(V )
φ(g1g2) = φ(g1)φ(g2).
V is called a representation space for φ.
For a fixed choice of basis GL(V ) ∼= GLn(C).
dimension of φ := dimension of V .
Notation: (φ,V ) or φ or V .
Note: For all g ∈ G , φ(g)V = V .
Pooja Singla (BGU) Representation Theory February 28, 2011 4 / 37
Basic Definitions
G - Always finite group.
Definition
A representation of G is a homomorphism from G to the set ofautomorphisms of a finite dimensional complex vector space V , i.e.
φ : G 7→ GL(V )
φ(g1g2) = φ(g1)φ(g2).
V is called a representation space for φ.
For a fixed choice of basis GL(V ) ∼= GLn(C).
dimension of φ := dimension of V .
Notation: (φ,V ) or φ or V .
Note: For all g ∈ G , φ(g)V = V .
Pooja Singla (BGU) Representation Theory February 28, 2011 4 / 37
Basic Definitions
G - Always finite group.
Definition
A representation of G is a homomorphism from G to the set ofautomorphisms of a finite dimensional complex vector space V , i.e.
φ : G 7→ GL(V )
φ(g1g2) = φ(g1)φ(g2).
V is called a representation space for φ.
For a fixed choice of basis GL(V ) ∼= GLn(C).
dimension of φ := dimension of V .
Notation: (φ,V ) or φ or V .
Note: For all g ∈ G , φ(g)V = V .
Pooja Singla (BGU) Representation Theory February 28, 2011 4 / 37
Basic Definitions
G - Always finite group.
Definition
A representation of G is a homomorphism from G to the set ofautomorphisms of a finite dimensional complex vector space V , i.e.
φ : G 7→ GL(V )
φ(g1g2) = φ(g1)φ(g2).
V is called a representation space for φ.
For a fixed choice of basis GL(V ) ∼= GLn(C).
dimension of φ := dimension of V .
Notation: (φ,V ) or φ or V .
Note: For all g ∈ G , φ(g)V = V .
Pooja Singla (BGU) Representation Theory February 28, 2011 4 / 37
Basic Definitions
G - Always finite group.
Definition
A representation of G is a homomorphism from G to the set ofautomorphisms of a finite dimensional complex vector space V , i.e.
φ : G 7→ GL(V )
φ(g1g2) = φ(g1)φ(g2).
V is called a representation space for φ.
For a fixed choice of basis GL(V ) ∼= GLn(C).
dimension of φ := dimension of V .
Notation: (φ,V ) or φ or V .
Note: For all g ∈ G , φ(g)V = V .
Pooja Singla (BGU) Representation Theory February 28, 2011 4 / 37
Basic Definitions
G - Always finite group.
Definition
A representation of G is a homomorphism from G to the set ofautomorphisms of a finite dimensional complex vector space V , i.e.
φ : G 7→ GL(V )
φ(g1g2) = φ(g1)φ(g2).
V is called a representation space for φ.
For a fixed choice of basis GL(V ) ∼= GLn(C).
dimension of φ := dimension of V .
Notation: (φ,V ) or φ or V .
Note: For all g ∈ G , φ(g)V = V .
Pooja Singla (BGU) Representation Theory February 28, 2011 4 / 37
Examples:
For any group G , the map φ : G → C?, given by φ(g) = 1 for all g .dim(φ) = 1.
Let Cn - Cyclic group with n elements. Let φ : Cn → C? ahomomorphism, then
φ(x)n = 1 for all x ∈ Cn.
Let ω be a generator of Cn and ζn be n− th primitive root of 1. Thenφ is completely determined by φ(ω).
Let φ(ω) = ζ in then we denote φ by φi .
Pooja Singla (BGU) Representation Theory February 28, 2011 5 / 37
Examples:
For any group G , the map φ : G → C?, given by φ(g) = 1 for all g .dim(φ) = 1.
Let Cn - Cyclic group with n elements. Let φ : Cn → C? ahomomorphism, then
φ(x)n = 1 for all x ∈ Cn.
Let ω be a generator of Cn and ζn be n− th primitive root of 1. Thenφ is completely determined by φ(ω).
Let φ(ω) = ζ in then we denote φ by φi .
Pooja Singla (BGU) Representation Theory February 28, 2011 5 / 37
Examples:
For any group G , the map φ : G → C?, given by φ(g) = 1 for all g .dim(φ) = 1.
Let Cn - Cyclic group with n elements. Let φ : Cn → C? ahomomorphism, then
φ(x)n = 1 for all x ∈ Cn.
Let ω be a generator of Cn and ζn be n− th primitive root of 1. Thenφ is completely determined by φ(ω).
Let φ(ω) = ζ in then we denote φ by φi .
Pooja Singla (BGU) Representation Theory February 28, 2011 5 / 37
Examples
Let S3 - permutations of {1, 2, 3}. Define φ : S3 → GL3(C) by
(1)→
1 0 00 1 00 0 1
, (12)→
0 1 01 0 00 0 1
(13)→
0 0 10 1 01 0 0
, (23)→
1 0 00 0 10 1 0
(123)→
0 0 11 0 00 1 0
, (132)→
0 1 00 0 11 0 0
.dim(φ) = 3.
Pooja Singla (BGU) Representation Theory February 28, 2011 6 / 37
Examples
(Permutation Representation) Suppose G acts on finite set X , that isfor each s ∈ G , there is given a permutation x 7→ sx of X satisfying
1x = x , s(tx) = (st)x , s, t ∈ G , x ∈ X .
Let V be complex vector space with basis (ex)x∈X . For s ∈ G , let
ρ : G → GL(V );
ρ(s) : ex 7→ esx .
dim(ρ) = |X |.(Regular Representation) If V is space with basis (eg )g∈G , then aboveaction is called regular representation of G .
Pooja Singla (BGU) Representation Theory February 28, 2011 7 / 37
Examples
(Permutation Representation) Suppose G acts on finite set X , that isfor each s ∈ G , there is given a permutation x 7→ sx of X satisfying
1x = x , s(tx) = (st)x , s, t ∈ G , x ∈ X .
Let V be complex vector space with basis (ex)x∈X . For s ∈ G , let
ρ : G → GL(V );
ρ(s) : ex 7→ esx .
dim(ρ) = |X |.
(Regular Representation) If V is space with basis (eg )g∈G , then aboveaction is called regular representation of G .
Pooja Singla (BGU) Representation Theory February 28, 2011 7 / 37
Examples
(Permutation Representation) Suppose G acts on finite set X , that isfor each s ∈ G , there is given a permutation x 7→ sx of X satisfying
1x = x , s(tx) = (st)x , s, t ∈ G , x ∈ X .
Let V be complex vector space with basis (ex)x∈X . For s ∈ G , let
ρ : G → GL(V );
ρ(s) : ex 7→ esx .
dim(ρ) = |X |.(Regular Representation) If V is space with basis (eg )g∈G , then aboveaction is called regular representation of G .
Pooja Singla (BGU) Representation Theory February 28, 2011 7 / 37
Question
Question
What are all the complex representations of a finite group G?
Pooja Singla (BGU) Representation Theory February 28, 2011 8 / 37
Tools
Definition
(G-invariant Space) A space V is called G -invariant if there existsφ : G → Aut(V ) such that
φ(g)V = V for all g ∈ G .
G -invariant ↔ G -representation
(Subrepresentation) Any G invariant subspace of V is calledsubrepresentation.
(Irreducible Representation) A representation is called irreducible if ithas no proper subrepresentations.
One dimensional representations are irreducible.
Pooja Singla (BGU) Representation Theory February 28, 2011 9 / 37
Tools
Definition
(G-invariant Space) A space V is called G -invariant if there existsφ : G → Aut(V ) such that
φ(g)V = V for all g ∈ G .
G -invariant ↔ G -representation
(Subrepresentation) Any G invariant subspace of V is calledsubrepresentation.
(Irreducible Representation) A representation is called irreducible if ithas no proper subrepresentations.
One dimensional representations are irreducible.
Pooja Singla (BGU) Representation Theory February 28, 2011 9 / 37
Tools
Definition
(G-invariant Space) A space V is called G -invariant if there existsφ : G → Aut(V ) such that
φ(g)V = V for all g ∈ G .
G -invariant ↔ G -representation
(Subrepresentation) Any G invariant subspace of V is calledsubrepresentation.
(Irreducible Representation) A representation is called irreducible if ithas no proper subrepresentations.
One dimensional representations are irreducible.
Pooja Singla (BGU) Representation Theory February 28, 2011 9 / 37
Tools
Definition
(G-invariant Space) A space V is called G -invariant if there existsφ : G → Aut(V ) such that
φ(g)V = V for all g ∈ G .
G -invariant ↔ G -representation
(Subrepresentation) Any G invariant subspace of V is calledsubrepresentation.
(Irreducible Representation) A representation is called irreducible if ithas no proper subrepresentations.
One dimensional representations are irreducible.
Pooja Singla (BGU) Representation Theory February 28, 2011 9 / 37
Tools
Definition
(G-invariant Space) A space V is called G -invariant if there existsφ : G → Aut(V ) such that
φ(g)V = V for all g ∈ G .
G -invariant ↔ G -representation
(Subrepresentation) Any G invariant subspace of V is calledsubrepresentation.
(Irreducible Representation) A representation is called irreducible if ithas no proper subrepresentations.
One dimensional representations are irreducible.
Pooja Singla (BGU) Representation Theory February 28, 2011 9 / 37
Tools
Definition
Two representations (φ1,V1) and (φ2,V2) are said to be equivalent ifthere exists an isomorphism T : V1 7→ V2 such that φ2(g)T = Tφ1(g) forall g ∈ G .
V1
T��
φ1(g) // V1
T��
V2φ2(g) // V2
In matrix notations: If dim(V1) = dim(V2) = n. Then(φ1,V1) ∼= (φ2,V2), if there exists X ∈ GLn(C) such that
Xφ1(g)X−1 = φ2(g).
Pooja Singla (BGU) Representation Theory February 28, 2011 10 / 37
Tools
Definition
Two representations (φ1,V1) and (φ2,V2) are said to be equivalent ifthere exists an isomorphism T : V1 7→ V2 such that φ2(g)T = Tφ1(g) forall g ∈ G .
V1
T��
φ1(g) // V1
T��
V2φ2(g) // V2
In matrix notations: If dim(V1) = dim(V2) = n. Then(φ1,V1) ∼= (φ2,V2), if there exists X ∈ GLn(C) such that
Xφ1(g)X−1 = φ2(g).
Pooja Singla (BGU) Representation Theory February 28, 2011 10 / 37
Tools
Definition
Two representations (φ1,V1) and (φ2,V2) are said to be equivalent ifthere exists an isomorphism T : V1 7→ V2 such that φ2(g)T = Tφ1(g) forall g ∈ G .
V1
T��
φ1(g) // V1
T��
V2φ2(g) // V2
In matrix notations: If dim(V1) = dim(V2) = n. Then(φ1,V1) ∼= (φ2,V2), if there exists X ∈ GLn(C) such that
Xφ1(g)X−1 = φ2(g).
Pooja Singla (BGU) Representation Theory February 28, 2011 10 / 37
Examples
Consider φ1 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[0 11 0
].
and φ2 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[1 00 −1
].
Claim: φ1∼= φ2.
Take X =
[1 11 −1
], then
X
[0 11 0
]X−1 =
[1 00 −1
].
Pooja Singla (BGU) Representation Theory February 28, 2011 11 / 37
Examples
Consider φ1 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[0 11 0
].
and φ2 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[1 00 −1
].
Claim: φ1∼= φ2.
Take X =
[1 11 −1
], then
X
[0 11 0
]X−1 =
[1 00 −1
].
Pooja Singla (BGU) Representation Theory February 28, 2011 11 / 37
Examples
Consider φ1 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[0 11 0
].
and φ2 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[1 00 −1
].
Claim: φ1∼= φ2.
Take X =
[1 11 −1
], then
X
[0 11 0
]X−1 =
[1 00 −1
].
Pooja Singla (BGU) Representation Theory February 28, 2011 11 / 37
Examples
Consider φ1 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[0 11 0
].
and φ2 : S2 → GL2(C) by
(1) 7→[
1 00 1
], (12) 7→
[1 00 −1
].
Claim: φ1∼= φ2.
Take X =
[1 11 −1
], then
X
[0 11 0
]X−1 =
[1 00 −1
].
Pooja Singla (BGU) Representation Theory February 28, 2011 11 / 37
Examples
Consider one dimensional representations φi and φj of cyclic group Cn
given byφi (ω) = ζ i
n , φj(ω) = ζ jn
where ω is generator of Cn.
Claim: For i 6= j , φi � φj .
Proof:If f : C→ C is an isomorphism then f (x) = λx for someλ ∈ C?.
Cλ
��
ζ in // C
�
Cζ jn // C
φi∼= φj implies λζ i
n = ζ jnλ, which is not true.
Pooja Singla (BGU) Representation Theory February 28, 2011 12 / 37
Examples
Consider one dimensional representations φi and φj of cyclic group Cn
given byφi (ω) = ζ i
n , φj(ω) = ζ jn
where ω is generator of Cn.
Claim: For i 6= j , φi � φj .
Proof:If f : C→ C is an isomorphism then f (x) = λx for someλ ∈ C?.
Cλ
��
ζ in // C
�
Cζ jn // C
φi∼= φj implies λζ i
n = ζ jnλ, which is not true.
Pooja Singla (BGU) Representation Theory February 28, 2011 12 / 37
Examples
Consider one dimensional representations φi and φj of cyclic group Cn
given byφi (ω) = ζ i
n , φj(ω) = ζ jn
where ω is generator of Cn.
Claim: For i 6= j , φi � φj .
Proof:If f : C→ C is an isomorphism then f (x) = λx for someλ ∈ C?.
Cλ
��
ζ in // C
�
Cζ jn // C
φi∼= φj implies λζ i
n = ζ jnλ, which is not true.
Pooja Singla (BGU) Representation Theory February 28, 2011 12 / 37
Definition
(G -linear map) Let (φ1,V1) and (φ2,V2) be two representations of finitegroup G . Then a map T : V1 → V2 is called G-linear if
1 T is C-linear.
2 T ◦ φ1(g) = φ2(g) ◦ T .
Lemma
The kernel and image of G -linear map are G -invariant subspaces.
Proof:
Let W1 ⊂ V1 be the kernel of T .
Take v ∈W1, we prove φ1(g)v ∈W1 for all g ∈ G .
T (φ1(g)v) = φ2(g)(T (v)) = 0.
φ1(g)v ∈W1.
Similar argument for the image.
Pooja Singla (BGU) Representation Theory February 28, 2011 13 / 37
Definition
(G -linear map) Let (φ1,V1) and (φ2,V2) be two representations of finitegroup G . Then a map T : V1 → V2 is called G-linear if
1 T is C-linear.
2 T ◦ φ1(g) = φ2(g) ◦ T .
Lemma
The kernel and image of G -linear map are G -invariant subspaces.
Proof:
Let W1 ⊂ V1 be the kernel of T .
Take v ∈W1, we prove φ1(g)v ∈W1 for all g ∈ G .
T (φ1(g)v) = φ2(g)(T (v)) = 0.
φ1(g)v ∈W1.
Similar argument for the image.
Pooja Singla (BGU) Representation Theory February 28, 2011 13 / 37
Definition
(G -linear map) Let (φ1,V1) and (φ2,V2) be two representations of finitegroup G . Then a map T : V1 → V2 is called G-linear if
1 T is C-linear.
2 T ◦ φ1(g) = φ2(g) ◦ T .
Lemma
The kernel and image of G -linear map are G -invariant subspaces.
Proof:
Let W1 ⊂ V1 be the kernel of T .
Take v ∈W1, we prove φ1(g)v ∈W1 for all g ∈ G .
T (φ1(g)v) = φ2(g)(T (v)) = 0.
φ1(g)v ∈W1.
Similar argument for the image.
Pooja Singla (BGU) Representation Theory February 28, 2011 13 / 37
Definition
(G -linear map) Let (φ1,V1) and (φ2,V2) be two representations of finitegroup G . Then a map T : V1 → V2 is called G-linear if
1 T is C-linear.
2 T ◦ φ1(g) = φ2(g) ◦ T .
Lemma
The kernel and image of G -linear map are G -invariant subspaces.
Proof:
Let W1 ⊂ V1 be the kernel of T .
Take v ∈W1, we prove φ1(g)v ∈W1 for all g ∈ G .
T (φ1(g)v) = φ2(g)(T (v)) = 0.
φ1(g)v ∈W1.
Similar argument for the image.
Pooja Singla (BGU) Representation Theory February 28, 2011 13 / 37
Definition
(G -linear map) Let (φ1,V1) and (φ2,V2) be two representations of finitegroup G . Then a map T : V1 → V2 is called G-linear if
1 T is C-linear.
2 T ◦ φ1(g) = φ2(g) ◦ T .
Lemma
The kernel and image of G -linear map are G -invariant subspaces.
Proof:
Let W1 ⊂ V1 be the kernel of T .
Take v ∈W1, we prove φ1(g)v ∈W1 for all g ∈ G .
T (φ1(g)v) = φ2(g)(T (v)) = 0.
φ1(g)v ∈W1.
Similar argument for the image.
Pooja Singla (BGU) Representation Theory February 28, 2011 13 / 37
Definition
(G -linear map) Let (φ1,V1) and (φ2,V2) be two representations of finitegroup G . Then a map T : V1 → V2 is called G-linear if
1 T is C-linear.
2 T ◦ φ1(g) = φ2(g) ◦ T .
Lemma
The kernel and image of G -linear map are G -invariant subspaces.
Proof:
Let W1 ⊂ V1 be the kernel of T .
Take v ∈W1, we prove φ1(g)v ∈W1 for all g ∈ G .
T (φ1(g)v) = φ2(g)(T (v)) = 0.
φ1(g)v ∈W1.
Similar argument for the image.
Pooja Singla (BGU) Representation Theory February 28, 2011 13 / 37
Definition
(G -linear map) Let (φ1,V1) and (φ2,V2) be two representations of finitegroup G . Then a map T : V1 → V2 is called G-linear if
1 T is C-linear.
2 T ◦ φ1(g) = φ2(g) ◦ T .
Lemma
The kernel and image of G -linear map are G -invariant subspaces.
Proof:
Let W1 ⊂ V1 be the kernel of T .
Take v ∈W1, we prove φ1(g)v ∈W1 for all g ∈ G .
T (φ1(g)v) = φ2(g)(T (v)) = 0.
φ1(g)v ∈W1.
Similar argument for the image.
Pooja Singla (BGU) Representation Theory February 28, 2011 13 / 37
More Tools
Definition
(Direct Sum of Representations) If (φ,V ) and (ψ,W ) be tworepresentations of group G , then (φ⊕ ψ,V ⊕W ) given by
[(φ⊕ ψ)(g)](v ,w) = (φ(g)v , ψ(g)w)
is a representation of G .
Observe dim(φ⊕ ψ) = dim(φ) + dim(ψ).
In terms of matrices
(φ⊕ ψ)(g) =
[φ(g) 0
0 ψ(g)
]
Pooja Singla (BGU) Representation Theory February 28, 2011 14 / 37
More Tools
Definition
(Direct Sum of Representations) If (φ,V ) and (ψ,W ) be tworepresentations of group G , then (φ⊕ ψ,V ⊕W ) given by
[(φ⊕ ψ)(g)](v ,w) = (φ(g)v , ψ(g)w)
is a representation of G .
Observe dim(φ⊕ ψ) = dim(φ) + dim(ψ).
In terms of matrices
(φ⊕ ψ)(g) =
[φ(g) 0
0 ψ(g)
]
Pooja Singla (BGU) Representation Theory February 28, 2011 14 / 37
More Tools
Definition
(Direct Sum of Representations) If (φ,V ) and (ψ,W ) be tworepresentations of group G , then (φ⊕ ψ,V ⊕W ) given by
[(φ⊕ ψ)(g)](v ,w) = (φ(g)v , ψ(g)w)
is a representation of G .
Observe dim(φ⊕ ψ) = dim(φ) + dim(ψ).
In terms of matrices
(φ⊕ ψ)(g) =
[φ(g) 0
0 ψ(g)
]
Pooja Singla (BGU) Representation Theory February 28, 2011 14 / 37
More Tools
Definition
(Direct Sum of Representations) If (φ,V ) and (ψ,W ) be tworepresentations of group G , then (φ⊕ ψ,V ⊕W ) given by
[(φ⊕ ψ)(g)](v ,w) = (φ(g)v , ψ(g)w)
is a representation of G .
Observe dim(φ⊕ ψ) = dim(φ) + dim(ψ).
In terms of matrices
(φ⊕ ψ)(g) =
[φ(g) 0
0 ψ(g)
]
Pooja Singla (BGU) Representation Theory February 28, 2011 14 / 37
Proposition
Let (φ,V ) be a complex representation of finite group G . The followingare equivalent :
1 (φ,V ) is irreducible.
2 (φ,V ) can not be written as direct sum of two propersubrepresentations.
Proof: Let W be a G -invariant subspace of V . For proof we show thatthere is a complimentary invariant subspace W ′ such that
V = W ⊕W ′.
Let U be an arbitrary complement of W in V , let
π0 : V →W
be the projection given by the direct sum decomposition V = W ⊕ U.
Pooja Singla (BGU) Representation Theory February 28, 2011 15 / 37
Proposition
Let (φ,V ) be a complex representation of finite group G . The followingare equivalent :
1 (φ,V ) is irreducible.
2 (φ,V ) can not be written as direct sum of two propersubrepresentations.
Proof: Let W be a G -invariant subspace of V . For proof we show thatthere is a complimentary invariant subspace W ′ such that
V = W ⊕W ′.
Let U be an arbitrary complement of W in V , let
π0 : V →W
be the projection given by the direct sum decomposition V = W ⊕ U.
Pooja Singla (BGU) Representation Theory February 28, 2011 15 / 37
Average the map π0 over G , that is an onto map π : V →W by,
π(v) =1
|G |∑g∈G
φ(g)(π0(φ(g)−1v)).
Then π is a G -linear. Therefore its kernel is the required G -invariantcomplement of W .
Pooja Singla (BGU) Representation Theory February 28, 2011 16 / 37
Theorem
Every complex representation of finite group G is direct sum of irreduciblerepresentations.
Proof:(For cyclic group case) Let φ : Cn → GLm(C)- homomorphism.
Step 1. Every finite order complex matrix is diagonalizable.
Step 2. The matrices φ(x) are pairwise commuting and diagonalizable,hence are simultaneously diagonalizable. Therefore φ is easily seen to bedirect sum of one dimensional representations. That is
φ = ⊕iφ⊕mii
where mi is the multiplicity.
General Proof Decompose V into irreducible representation by usinglast proposition.
Pooja Singla (BGU) Representation Theory February 28, 2011 17 / 37
Theorem
Every complex representation of finite group G is direct sum of irreduciblerepresentations.
Proof:(For cyclic group case) Let φ : Cn → GLm(C)- homomorphism.
Step 1. Every finite order complex matrix is diagonalizable.
Step 2. The matrices φ(x) are pairwise commuting and diagonalizable,hence are simultaneously diagonalizable. Therefore φ is easily seen to bedirect sum of one dimensional representations. That is
φ = ⊕iφ⊕mii
where mi is the multiplicity.
General Proof Decompose V into irreducible representation by usinglast proposition.
Pooja Singla (BGU) Representation Theory February 28, 2011 17 / 37
Theorem
Every complex representation of finite group G is direct sum of irreduciblerepresentations.
Proof:(For cyclic group case) Let φ : Cn → GLm(C)- homomorphism.
Step 1. Every finite order complex matrix is diagonalizable.
Step 2. The matrices φ(x) are pairwise commuting and diagonalizable,hence are simultaneously diagonalizable. Therefore φ is easily seen to bedirect sum of one dimensional representations. That is
φ = ⊕iφ⊕mii
where mi is the multiplicity.
General Proof Decompose V into irreducible representation by usinglast proposition.
Pooja Singla (BGU) Representation Theory February 28, 2011 17 / 37
Theorem
Every complex representation of finite group G is direct sum of irreduciblerepresentations.
Proof:(For cyclic group case) Let φ : Cn → GLm(C)- homomorphism.
Step 1. Every finite order complex matrix is diagonalizable.
Step 2. The matrices φ(x) are pairwise commuting and diagonalizable,hence are simultaneously diagonalizable. Therefore φ is easily seen to bedirect sum of one dimensional representations. That is
φ = ⊕iφ⊕mii
where mi is the multiplicity.
General Proof Decompose V into irreducible representation by usinglast proposition.
Pooja Singla (BGU) Representation Theory February 28, 2011 17 / 37
Theorem
Every complex representation of finite group G is direct sum of irreduciblerepresentations.
Proof:(For cyclic group case) Let φ : Cn → GLm(C)- homomorphism.
Step 1. Every finite order complex matrix is diagonalizable.
Step 2. The matrices φ(x) are pairwise commuting and diagonalizable,hence are simultaneously diagonalizable. Therefore φ is easily seen to bedirect sum of one dimensional representations. That is
φ = ⊕iφ⊕mii
where mi is the multiplicity.
General Proof Decompose V into irreducible representation by usinglast proposition.
Pooja Singla (BGU) Representation Theory February 28, 2011 17 / 37
So the question is...
Question
What are all the finite dimensional inequivalent irreducible complexrepresentations of a given finite group G ?
Observe we have answered it already for cyclic groups.
Pooja Singla (BGU) Representation Theory February 28, 2011 18 / 37
So the question is...
Question
What are all the finite dimensional inequivalent irreducible complexrepresentations of a given finite group G ?
Observe we have answered it already for cyclic groups.
Pooja Singla (BGU) Representation Theory February 28, 2011 18 / 37
More tools and Interesting Results
Theorem
(Schur’s Lemma) Let φ1 : G → GL(V1) and φ2 : G → GL(V2) be twoirreducible representations of G , and let T be a linear mapping of V1 intoV2 such that φ2(g) ◦ T = T ◦ φ1(g) for all g ∈ G . Then :
1 If φ1 and φ2 are not isomorphic, we have T = 0.
2 If V1 = V2 and φ1 = φ2, then T (x) = λx for x ∈ V and for somescalar λ ∈ C.
Proof:
Suppose T 6= 0. The G -linearity of T implies that both kernel andimage of T are G -invariant subspaces.
Pooja Singla (BGU) Representation Theory February 28, 2011 19 / 37
More tools and Interesting Results
Theorem
(Schur’s Lemma) Let φ1 : G → GL(V1) and φ2 : G → GL(V2) be twoirreducible representations of G , and let T be a linear mapping of V1 intoV2 such that φ2(g) ◦ T = T ◦ φ1(g) for all g ∈ G . Then :
1 If φ1 and φ2 are not isomorphic, we have T = 0.
2 If V1 = V2 and φ1 = φ2, then T (x) = λx for x ∈ V and for somescalar λ ∈ C.
Proof:
Suppose T 6= 0. The G -linearity of T implies that both kernel andimage of T are G -invariant subspaces.
Pooja Singla (BGU) Representation Theory February 28, 2011 19 / 37
More tools and Interesting Results
Theorem
(Schur’s Lemma) Let φ1 : G → GL(V1) and φ2 : G → GL(V2) be twoirreducible representations of G , and let T be a linear mapping of V1 intoV2 such that φ2(g) ◦ T = T ◦ φ1(g) for all g ∈ G . Then :
1 If φ1 and φ2 are not isomorphic, we have T = 0.
2 If V1 = V2 and φ1 = φ2, then T (x) = λx for x ∈ V and for somescalar λ ∈ C.
Proof:
Suppose T 6= 0. The G -linearity of T implies that both kernel andimage of T are G -invariant subspaces.
Pooja Singla (BGU) Representation Theory February 28, 2011 19 / 37
More interesting tools...
The irreduciblity of φ1 and φ2 implies T is an isomorphism.
If V1 = V2, φ1 = φ2. Let λ be an eigenvalue of T (recall field is C).
Put T ′ = T − λ. Then Ker(T ′) 6= 0.
Also φ2(g) ◦ T ′ = T ′ ◦ φ1(g).
By the first part, we must have T ′ = 0. That is T = λ.
Theorem
The number of inequivalent irreducible representations of finite group isequal to the number of its conjugacy classes.
Pooja Singla (BGU) Representation Theory February 28, 2011 20 / 37
More interesting tools...
The irreduciblity of φ1 and φ2 implies T is an isomorphism.
If V1 = V2, φ1 = φ2. Let λ be an eigenvalue of T (recall field is C).
Put T ′ = T − λ. Then Ker(T ′) 6= 0.
Also φ2(g) ◦ T ′ = T ′ ◦ φ1(g).
By the first part, we must have T ′ = 0. That is T = λ.
Theorem
The number of inequivalent irreducible representations of finite group isequal to the number of its conjugacy classes.
Pooja Singla (BGU) Representation Theory February 28, 2011 20 / 37
More interesting tools...
The irreduciblity of φ1 and φ2 implies T is an isomorphism.
If V1 = V2, φ1 = φ2. Let λ be an eigenvalue of T (recall field is C).
Put T ′ = T − λ. Then Ker(T ′) 6= 0.
Also φ2(g) ◦ T ′ = T ′ ◦ φ1(g).
By the first part, we must have T ′ = 0. That is T = λ.
Theorem
The number of inequivalent irreducible representations of finite group isequal to the number of its conjugacy classes.
Pooja Singla (BGU) Representation Theory February 28, 2011 20 / 37
More interesting tools...
The irreduciblity of φ1 and φ2 implies T is an isomorphism.
If V1 = V2, φ1 = φ2. Let λ be an eigenvalue of T (recall field is C).
Put T ′ = T − λ. Then Ker(T ′) 6= 0.
Also φ2(g) ◦ T ′ = T ′ ◦ φ1(g).
By the first part, we must have T ′ = 0. That is T = λ.
Theorem
The number of inequivalent irreducible representations of finite group isequal to the number of its conjugacy classes.
Pooja Singla (BGU) Representation Theory February 28, 2011 20 / 37
More interesting tools...
The irreduciblity of φ1 and φ2 implies T is an isomorphism.
If V1 = V2, φ1 = φ2. Let λ be an eigenvalue of T (recall field is C).
Put T ′ = T − λ. Then Ker(T ′) 6= 0.
Also φ2(g) ◦ T ′ = T ′ ◦ φ1(g).
By the first part, we must have T ′ = 0. That is T = λ.
Theorem
The number of inequivalent irreducible representations of finite group isequal to the number of its conjugacy classes.
Pooja Singla (BGU) Representation Theory February 28, 2011 20 / 37
More interesting tools...
The irreduciblity of φ1 and φ2 implies T is an isomorphism.
If V1 = V2, φ1 = φ2. Let λ be an eigenvalue of T (recall field is C).
Put T ′ = T − λ. Then Ker(T ′) 6= 0.
Also φ2(g) ◦ T ′ = T ′ ◦ φ1(g).
By the first part, we must have T ′ = 0. That is T = λ.
Theorem
The number of inequivalent irreducible representations of finite group isequal to the number of its conjugacy classes.
Pooja Singla (BGU) Representation Theory February 28, 2011 20 / 37
More tools and Interesting Results
Theorem
If ρ1, ρ2, . . . , ρt are all the inequivalent irreducible representations of groupG then
|G | =t∑
i=1
dim(ρi )2.
Consider the group S3.
|S3| = 6.
S3 has three conjugacy classes given by (1), (12), (123).
Define σ : S3 → C? by
σ(1) = 1, σ(123) = σ(132) = 1,
σ(12) = σ(23) = σ(13) = −1.
Pooja Singla (BGU) Representation Theory February 28, 2011 21 / 37
More tools and Interesting Results
Theorem
If ρ1, ρ2, . . . , ρt are all the inequivalent irreducible representations of groupG then
|G | =t∑
i=1
dim(ρi )2.
Consider the group S3.
|S3| = 6.
S3 has three conjugacy classes given by (1), (12), (123).
Define σ : S3 → C? by
σ(1) = 1, σ(123) = σ(132) = 1,
σ(12) = σ(23) = σ(13) = −1.
Pooja Singla (BGU) Representation Theory February 28, 2011 21 / 37
σ and trivial representations are two one dimensional inequivalentirreducible representations of S3 of dimension one.
The only way to write 6 as sum of three squares is 6 = 1 + 1 + 22.
Recall that the permutation representation of S3 which maps eachpermutation to corresponding permutation matrices.
This is not direct sum of one dimensional representations.
The only decomposition possible is 3 = 1 + 2.
So its decomposition with above observations will give all theirreducible representations of S3.
Pooja Singla (BGU) Representation Theory February 28, 2011 22 / 37
Permutation Representation
Let S3 - permutations of {1, 2, 3}. Define φ : S3 → GL3(C) by
(1)→
1 0 00 1 00 0 1
, (12)→
0 1 01 0 00 0 1
(13)→
0 0 10 1 01 0 0
, (23)→
1 0 00 0 10 1 0
(123)→
0 0 11 0 00 1 0
, (132)→
0 1 00 0 11 0 0
.dim(φ) = 3.
Pooja Singla (BGU) Representation Theory February 28, 2011 23 / 37
Groups GL2(K )
Set G = GL2(K ), |K | = q where q = pr for an odd prime p.
Question
What are all the irreducible complex representations of G ?
We shall construct a very special class of representations of these groups.
Pooja Singla (BGU) Representation Theory February 28, 2011 24 / 37
Remark
A representation of dimension one of G is simply a homomorphism φ of Gto C?.
Let G0 = {xyx−1y−1 | x , y ∈ G} be the derived subgroup of G .
Let π : G → G/G0 be the natural epimorphism.
Proposition
(One dimensional representations of G) Let φ : G → C? be ahomomorphism.
Then there exists a homomorphism φ : G/G0 → C? such thatφ.π = φ.
Conversely, any homomorphism from G/G0 to C? produces ahomomorphism from G to C? by composition with π.
Pooja Singla (BGU) Representation Theory February 28, 2011 25 / 37
Remark
A representation of dimension one of G is simply a homomorphism φ of Gto C?.
Let G0 = {xyx−1y−1 | x , y ∈ G} be the derived subgroup of G .
Let π : G → G/G0 be the natural epimorphism.
Proposition
(One dimensional representations of G) Let φ : G → C? be ahomomorphism.
Then there exists a homomorphism φ : G/G0 → C? such thatφ.π = φ.
Conversely, any homomorphism from G/G0 to C? produces ahomomorphism from G to C? by composition with π.
Pooja Singla (BGU) Representation Theory February 28, 2011 25 / 37
Remark
A representation of dimension one of G is simply a homomorphism φ of Gto C?.
Let G0 = {xyx−1y−1 | x , y ∈ G} be the derived subgroup of G .
Let π : G → G/G0 be the natural epimorphism.
Proposition
(One dimensional representations of G) Let φ : G → C? be ahomomorphism.
Then there exists a homomorphism φ : G/G0 → C? such thatφ.π = φ.
Conversely, any homomorphism from G/G0 to C? produces ahomomorphism from G to C? by composition with π.
Pooja Singla (BGU) Representation Theory February 28, 2011 25 / 37
Definition
(Induced Representations) Let H be a subgroup of a finite group G, andlet (ψ,U) be a representation of H. Let
V = {f : G → U|f (hg) = ψ(h)f (g), h ∈ H, g ∈ G}.
Then G acts on V by right translations; φ : G → Aut(V ) by
[φ(g)(f )](g ′) = f (g ′g) g , g ′ ∈ G , f ∈ V .
It follows that (φ,V ) is a representation of G . This is called inducedrepresentation of (ψ,U) from H to G , denoted by IndG
H(ψ).
dim(IndGH(ψ)) = |G/H|.dim(ψ).
By restricting the action of φ on a subgroup H of G , we obtain arepresentation of H, denoted by ResGH(φ).
Pooja Singla (BGU) Representation Theory February 28, 2011 26 / 37
Definition
(Induced Representations) Let H be a subgroup of a finite group G, andlet (ψ,U) be a representation of H. Let
V = {f : G → U|f (hg) = ψ(h)f (g), h ∈ H, g ∈ G}.
Then G acts on V by right translations; φ : G → Aut(V ) by
[φ(g)(f )](g ′) = f (g ′g) g , g ′ ∈ G , f ∈ V .
It follows that (φ,V ) is a representation of G . This is called inducedrepresentation of (ψ,U) from H to G , denoted by IndG
H(ψ).
dim(IndGH(ψ)) = |G/H|.dim(ψ).
By restricting the action of φ on a subgroup H of G , we obtain arepresentation of H, denoted by ResGH(φ).
Pooja Singla (BGU) Representation Theory February 28, 2011 26 / 37
Definition
(Induced Representations) Let H be a subgroup of a finite group G, andlet (ψ,U) be a representation of H. Let
V = {f : G → U|f (hg) = ψ(h)f (g), h ∈ H, g ∈ G}.
Then G acts on V by right translations; φ : G → Aut(V ) by
[φ(g)(f )](g ′) = f (g ′g) g , g ′ ∈ G , f ∈ V .
It follows that (φ,V ) is a representation of G . This is called inducedrepresentation of (ψ,U) from H to G , denoted by IndG
H(ψ).
dim(IndGH(ψ)) = |G/H|.dim(ψ).
By restricting the action of φ on a subgroup H of G , we obtain arepresentation of H, denoted by ResGH(φ).
Pooja Singla (BGU) Representation Theory February 28, 2011 26 / 37
Set G = GL2(K ), |K | = q where q = pr for an odd prime p.
Question
What are all the irreducible complex representations of G ?
Let K ∗ be invertible elements of K .
Recall K ? is a cyclic group of order q − 1.
Let K ? is the set of one dimensional representations of K ?.
Let
B = {(
a b0 c
): a, c ∈ K ∗, b ∈ K}.
Choose µ1, µ2 ∈ K ∗.Define
µ : B 7→ C?;
(a b0 c
)7→ µ1 (a)µ2 (c) .
Pooja Singla (BGU) Representation Theory February 28, 2011 27 / 37
Writeµ = (µ1, µ2).
SetVµ = {f : G → C | f (bg) = µ(b)f (g)}.
Now define an action of G on Vµ by
(µ(g)f )(x) = f (xg) , for all x , g ∈ G .
dimension of µ = |G ||B| = (q2−1)(q2−q)
(q−1)2q= q + 1.
Question
Is µ irreducible ?
Pooja Singla (BGU) Representation Theory February 28, 2011 28 / 37
Writeµ = (µ1, µ2).
SetVµ = {f : G → C | f (bg) = µ(b)f (g)}.
Now define an action of G on Vµ by
(µ(g)f )(x) = f (xg) , for all x , g ∈ G .
dimension of µ = |G ||B| = (q2−1)(q2−q)
(q−1)2q= q + 1.
Question
Is µ irreducible ?
Pooja Singla (BGU) Representation Theory February 28, 2011 28 / 37
Writeµ = (µ1, µ2).
SetVµ = {f : G → C | f (bg) = µ(b)f (g)}.
Now define an action of G on Vµ by
(µ(g)f )(x) = f (xg) , for all x , g ∈ G .
dimension of µ = |G ||B| = (q2−1)(q2−q)
(q−1)2q= q + 1.
Question
Is µ irreducible ?
Pooja Singla (BGU) Representation Theory February 28, 2011 28 / 37
Writeµ = (µ1, µ2).
SetVµ = {f : G → C | f (bg) = µ(b)f (g)}.
Now define an action of G on Vµ by
(µ(g)f )(x) = f (xg) , for all x , g ∈ G .
dimension of µ = |G ||B| = (q2−1)(q2−q)
(q−1)2q= q + 1.
Question
Is µ irreducible ?
Pooja Singla (BGU) Representation Theory February 28, 2011 28 / 37
Case 1. For µ1 6= µ2, the G -representation (µ,Vµ) is irreducible.Furthermore,
(µ1, µ2) ∼= (µ′1, µ′2)
if and only if either
µ1 = µ′1 and µ2 = µ
′2
orµ1 = µ
′2 and µ2 = µ
′1.
This gives (q−1)(q−2)2 inequivalent (q + 1) dimensional irreducible
representations.
Pooja Singla (BGU) Representation Theory February 28, 2011 29 / 37
Case 2. For µ1 = µ2, the G -representation (µ,Vµ) is not irreducible.In this case
µ = φ1 ⊕ φ2,
where φ1 is one dimensional and φ2 is q dimensional.
Suppose µ = φ1 ⊕ φ2 and ν = ψ1 ⊕ ψ2.If µ � ν then
φ1 � ψ1 and φ2 � ψ2.
This gives (q − 1) one dimensional representations and (q − 1)inequivalent q dimensional representations.
Pooja Singla (BGU) Representation Theory February 28, 2011 30 / 37
Table of Conjugacy classes
conjugacy class representative No. of classes
central semisimple
(a 00 a
)q − 1
unitary
(a 10 a
)q − 1
non central semisimple
(a 00 b
), a 6= b (q−1)(q−2)
2
anisotropic
(0 −αα1 α + α
)q2−q
2
Pooja Singla (BGU) Representation Theory February 28, 2011 31 / 37
Mackey’s intertwining Theorem
Let H and K be subgroups of G . If g ∈ G then the setHgK = {hgk | h ∈ H, k ∈ K} is called a double coset with respect tosubgroup H and K . The element g is called its representative.
A complete set of representatives of all (H,K )-double cosets isdenoted by H\G/K .
For s ∈ H\G/K we set Hs = sHs−1 ∩ K .
Consider the representation (τ,W ) of H.
The subgroup Hs has a natural representation (τ s ,W ) defined by
τ s(x) = τ(s−1xs), x ∈ Hs
Pooja Singla (BGU) Representation Theory February 28, 2011 32 / 37
If (φ,V ) and (ψ,W ) are two G -representations then
HomG (φ, ψ) = {T : V →W | T is G − linear}.
Theorem
(Mackey’s Intertwining Theorem) Let H and K be subgroups of G and(U, σ) a representation of K and (W , τ) a representation of H. Then:
HomG (IndGK (σ), IndG
H(τ)) ∼= ⊕s∈H\G/K
HomsHs−1∩K (σ, τ s).
Pooja Singla (BGU) Representation Theory February 28, 2011 33 / 37
Continuation for Cases I and II
For our case H = K = B, where B = {(
a b0 c
)| a, c ,∈ K ?, b ∈ K}
and σ = τ = (µ1, µ2).
For c 6= 0:(a bc d
)=
(b − ac−1d a
0 c
)(0 11 0
)(1 c−1d0 1
)The double coset representatives B\GL2(K )/B correspond to
{(
1 00 1
),
(0 11 0
)},
Let s =
(0 11 0
). Then
B ∩ sBs−1 = T = {(
a 00 b
)| a.b ∈ K ?}.
Pooja Singla (BGU) Representation Theory February 28, 2011 34 / 37
Continuation for Cases I and II
For our case H = K = B, where B = {(
a b0 c
)| a, c ,∈ K ?, b ∈ K}
and σ = τ = (µ1, µ2).For c 6= 0:(
a bc d
)=
(b − ac−1d a
0 c
)(0 11 0
)(1 c−1d0 1
)
The double coset representatives B\GL2(K )/B correspond to
{(
1 00 1
),
(0 11 0
)},
Let s =
(0 11 0
). Then
B ∩ sBs−1 = T = {(
a 00 b
)| a.b ∈ K ?}.
Pooja Singla (BGU) Representation Theory February 28, 2011 34 / 37
Continuation for Cases I and II
For our case H = K = B, where B = {(
a b0 c
)| a, c ,∈ K ?, b ∈ K}
and σ = τ = (µ1, µ2).For c 6= 0:(
a bc d
)=
(b − ac−1d a
0 c
)(0 11 0
)(1 c−1d0 1
)The double coset representatives B\GL2(K )/B correspond to
{(
1 00 1
),
(0 11 0
)},
Let s =
(0 11 0
). Then
B ∩ sBs−1 = T = {(
a 00 b
)| a.b ∈ K ?}.
Pooja Singla (BGU) Representation Theory February 28, 2011 34 / 37
Continuation for Cases I and II
For our case H = K = B, where B = {(
a b0 c
)| a, c ,∈ K ?, b ∈ K}
and σ = τ = (µ1, µ2).For c 6= 0:(
a bc d
)=
(b − ac−1d a
0 c
)(0 11 0
)(1 c−1d0 1
)The double coset representatives B\GL2(K )/B correspond to
{(
1 00 1
),
(0 11 0
)},
Let s =
(0 11 0
). Then
B ∩ sBs−1 = T = {(
a 00 b
)| a.b ∈ K ?}.
Pooja Singla (BGU) Representation Theory February 28, 2011 34 / 37
Observe that
(µ1, µ2)s
(x 00 y
)= µ2(x)µ1(y).
Hence by Mackey’s intertwining theorem we obtain that
HomG ( (µ1, µ2), (µ1, µ2)) = [HomB((µ1, µ2), (µ1, µ2))]
⊕[HomT ((µ1, µ2), (µ2, µ1))]
The T -representations (µ1, µ2) and (µ2, µ1) are equivalent if and only ifµ1 = µ2.
This implies dimC(HomG ( (µ1, µ2), (µ1, µ2))) is equal to one for µ1 6= µ2
and is equal to two for µ1 = µ2.
This combined with Schur’s Lemma gives the result.
Pooja Singla (BGU) Representation Theory February 28, 2011 35 / 37
Observe that
(µ1, µ2)s
(x 00 y
)= µ2(x)µ1(y).
Hence by Mackey’s intertwining theorem we obtain that
HomG ( (µ1, µ2), (µ1, µ2)) = [HomB((µ1, µ2), (µ1, µ2))]
⊕[HomT ((µ1, µ2), (µ2, µ1))]
The T -representations (µ1, µ2) and (µ2, µ1) are equivalent if and only ifµ1 = µ2.
This implies dimC(HomG ( (µ1, µ2), (µ1, µ2))) is equal to one for µ1 6= µ2
and is equal to two for µ1 = µ2.
This combined with Schur’s Lemma gives the result.
Pooja Singla (BGU) Representation Theory February 28, 2011 35 / 37
Summary
dimension No of irreducible namerepresentations
1 q − 1 one dimensional
q q − 1 special
q + 1 (q−1)(q−2)2 regular principal series
q − 1 q2−q2 cuspidal
Pooja Singla (BGU) Representation Theory February 28, 2011 36 / 37
References
Fulton, William and Harris, Joe,Representation theory,Graduate Texts in Mathematics, Springer-Verlag, 1991.
Serre, Jean-PierreLinear representations of finite groupsSpringer-Verlag 1977.
Ilya Piatetski-Shapirocomplex representations of GL(2,K ) for finite fields KContemporary Mathematics, 1983.
Aburto, Luisa and Johnson, Roberto and Pantoja, JoseeThe complex linear representations of GL(2, k), k a finite field,Proyecciones. Journal of Mathematics,Vol. 25, 2006, 307–329.
Pooja Singla (BGU) Representation Theory February 28, 2011 37 / 37